Impact of price cap regulation on supply chain contracting between two monopolists

Jing Feng; Yanfei Lan; Ruiqing Zhao

doi:10.3934/jimo.2016021

Article Contents

2017, Volume 13, Issue 1: 349-373. Doi: 10.3934/jimo.2016021

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Impact of price cap regulation on supply chain contracting between two monopolists

Institute of Systems Engineering, Tianjin University, Tianjin 300072, China

* Corresponding author: Yanfei Lan, Email: lanyf@tju.edu.cn
* Corresponding author: Yanfei Lan, Email: lanyf@tju.edu.cn

Received: July 31, 2013

Published: August 2017

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Abstract

This paper considers a supply chain with an unregulated upstream monopolist (she) supplying a kind of products to a regulated downstream monopolist (he). The upstream monopolist's production efficiency, which represents her type, is only privately known to herself. When the downstream monopolist trades with the upstream monopolist, his pricing discretion is constrained by price cap regulation (PCR). We model this problem as a game of adverse selection with the price cap constraint. In this model, the downstream monopolist offers a menu of contracts, each of which consists of two parameters: the transfer payment and the retail price. We show that private information can weaken PCR's impact on the optimal contract, and PCR can dampen the effects of private information. We also shed light on the influences of private information and PCR on the optimal contract, the downstream monopolist's profit, the upstream monopolist's profit, the consumers' surplus and the social total welfare, respectively. Finally, a numerical example is given to illustrate the proposed results.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Information structure's impact on the optimal contract

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Figure 2. Price cap's impact on the optimal contract under full information

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Figure 3. Price cap's impact on the revenue under full information

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Figure 4. Price cap's impact on the optimal contract under private information

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Figure 5. Price cap's impact on the revenue under private information

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Table 1. Information structure's impact on contract

	$Scenario$	$p$	$t$
$full$	$information$	$0.5x+13$	$-0.7x^{2}+6.5x+7$
$private$	$information$	$x+8$	$-0.5x^{2}-x+63.3$

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Table 2. Price cap's impact on the optimal contact under full information

$Scenario$	$p^{**}$	$t^{**}$
$RPCR$	$0.5x+13$	$-0.7x^{2}+6.5x+7$
$TPCR$	$16.75$	$-0.2x^{2}+3.25x+3.25$
$MPCR\,when\,x\geq x_{0}$	17.75	$-0.2x^{2}+2.25x+2.25$
$MPCR\,when\, x < x_{0}$	$0.5x+13$	$-0.7x^{2}+6.5x+7$

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Table 3. Price cap's impact on the optimal contact under private information

$Scenario$	$p_{1}^{**}$	$t_{1}^{**}$
$RPCR$	$x+8$	$-0.5x^{2}-x+63.3$
$TPCR $	$16.75$	16.3
$MPCR\, when\, x\geq x_{1}$	$17.75$	5.175
$MPCR\,when\, x< x_{1}$	$x+8$	$-0.5x^{2}-x+63.3$

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Table 4. Price cap's impacts on the downstream monopolist's profit, the consumers' surplus and the social total welfare under full information

$Scenario$	$U^{**}$	$S^{**}$	$W^{**}$
$RPCR$	$0.45x^{2}\!-\!7x+49$	$0.125x^{2}\!-\!3.5x\!+\!24.5$	$0.575x^{2}\!-\!10.5x\!+\!73.5$
$TPCR$	$0.2x^{2}\!-\!3.25x\!+\!34.94$	$5.28$	$0.2x^{2}\!-\!3.25x\!+\!40.22$
$MPCR\,when\,x\geq x_{0}$	$0.2x^{2}\!-\!2.25x\!+\!26.44$	$2.53$	$0.2x^{2}\!-\!2.25x\!+\!28.97$
$MPCR\,when\, x < x_{0}$	$0.45x^{2}\!-\!7x\!+\!49$	$0.125x^{2}\!-\!3.5x\!+\!24.5$	$0.575x^{2}\!-\!10.5x\!+\!73.5$

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Table 5. Price cap's impacts on the downstream monopolist's profit, the upstream monopolist's profit, the consumers surplus and the social total welfare under private information

$Scenario$	$U_{1}^{**}$	$\pi_{1}^{**}$	$S_{1}^{**}$	$W_{1}^{**}$
$RPCR$	$22.53$	$0.7x^{2}\!-\!12x\!+\!51.3$	$0.5x^{2}\!-\!12x\!+\!72$	$1.2x^{2}\!-\!24x\!+\!145.83$
$TPCR$	$21.89$	$0.2x^{2}\!-\!3.25x\!+\!13.05$	$5.28$	$0.2x^{2}\!-\!3,25x\!+\!40.22$
$MPCR\,when\, x\geq x_{1}$	$22.52$	$0.2x^{2}\!-\!2.25x\!+\!2.93$	2.53	$0.2x^{2}\!-\!2,25x\!+\!27.98$
$MPCR\,when\, x < x_{1}$	$22.52$	$0.7x^{2}\!-\!12x\!+\!51.3$	$0.5x^{2}\!-\!12x\!+\!72$	$1.2x^{2}\!-\!24x\!+\!145.82$

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